Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

bubokribuck · 2012-04-16 23:25:58

First I'm asked to expand the equation into a Taylor series and obtain the approximate solution for x=0.1 and x=0.2, and the answers I've calculated are: y(0.1)=0.109599 and y(0.2)=0.118396.

Next, I'm asked to write down the above problem as a system of first order differential equations and calculate the numerical solution at x = 0.2 using the Modified Euler method. Take the step-length h = 0.1 and work to 6 decimal places accuracy.

How can I rewrite the above equation as a first order DE? Is it

?

Last edited by bubokribuck (2012-04-17 02:12:22)

bobbym · 2012-04-17 00:25:28

Hi bubokribuck;

Can I see your Taylor expansion please? Also what accuracy were you asked for?

bubokribuck · 2012-04-17 02:31:05

Hi bobbym, thanks for your reply, this is the full question:

given
i) You are asked to find the approximate solution for this problem using the Taylor series method. Your expansion should include the first three non-zero terms and you should work to six decimal places accuracy. First find the approximate solutions for both y (0.1) and y(0.1) using the first three non-zero terms of Taylor series expansion for each function and then use this information to calculate the approximate solution at x = 0.2.
ii) Write down the above problem as a system of first order differential equations. Calculate the numerical solution at x = 0.2 using the Modified Euler method. Take the step-length h = 0.1 and work to 6 decimal places accuracy. Compare with your solution in part (i) and comment on your answers.

My taylor expansion is:

Last edited by bubokribuck (2012-04-17 02:32:23)

bobbym · 2012-04-17 05:03:12

Hi;

I am getting something else. You should get close to this.

y(.1)=0.109388

y(.2)=0.117511

Do you see your mistake?

bubokribuck · 2012-04-17 07:56:39

Um.... sorry, can't figure out what's wrong , is it that y''=/=-0.0802?

bobbym · 2012-04-17 08:00:04

Please take a closer look at

Presumably, you got that from here

bubokribuck · 2012-04-17 10:33:54

Oh sorry, I think I've typed it wrong, the equation should be

.
Rearrange the equation I get .

Last edited by bubokribuck (2012-04-17 10:35:58)

bobbym · 2012-04-17 10:48:21

Hi;

That changes everything. Okay, that is I recompute the answers and get what you got.

The exact answers are:

y(.1) = .1095798

y(.2) = .1182451

So you are close with only two terms.

bubokribuck · 2012-04-17 11:39:28

Phew, so that's taylor series done. XD
Thank you very much bobbym!

So now we have to rearrange the equation into a first order differential equation. This is the part that I'm not sure about. How can I make it a 1st order DE if it contains the term y''?

bubokribuck · 2012-04-17 11:46:18

Also I've ben given other equations such as

and and asked to write them in the form of first order differential equation and then use the Euler's Modified method to approximate the values at x=1.1.

I have no idea how to do it. Could you give me any hints please. Thanks!

Last edited by bubokribuck (2012-04-17 11:47:48)

bobbym · 2012-04-17 11:53:59

Hi bubokribuck;

Do you mean a system of linear first order DE's?

bubokribuck · 2012-04-17 12:07:33

I think so. The exact words from the question are "Write down the above problem as a system of first order differential equations". I'm not quite sure what it means. I've come across problems such as

so I know I can apply the Euler's method:

But those equations all contain the term y'', I've not come across that, so not sure what to do.

bobbym · 2012-04-17 12:15:25

Is that 2 separate problems in post #10?

bubokribuck · 2012-04-17 12:41:20

Yes. Those two equations in post #10 are from two other problems. There are 3 problems in total, and I need to use the modified Euler's method to find the numerical solutions:

bobbym · 2012-04-17 13:38:20

Hi bubokribuck;

For

you can form a system of 2 linear first order DE's.

With

bubokribuck · 2012-04-17 22:02:27

Hi bobbym,

Do I have to solve for p(x) first? How can I solve it please?

If

, do I write it as ?

Last edited by bubokribuck (2012-04-17 22:02:49)

bobbym · 2012-04-17 22:04:00

What formula do you use for the modified Euler?

bubokribuck · 2012-04-18 04:33:18

I use

Last edited by bubokribuck (2012-04-18 04:37:33)

bobbym · 2012-04-18 07:07:05

I use a slightly different one. As I understand there are a lot of these. Are you being restricted to one and only one formula?

mathmatiKs · 2012-04-18 12:23:35

I have a question.
In what year of school or college learned that?

bobbym · 2012-04-18 18:46:38

Hi mathmatiKs;

None, that I am aware of. It is taught in numerical analysis courses. Or it is learned the hard way.

Hi bubokribuck;

Want me to use one of my own? The one you provided is not for a system of equations.

bubokribuck · 2012-04-19 05:58:23

Hi bobbym,

If you can show your way of doing the question that'd be great. I was given these questions as some exercises for the test I'll be taking on Monday, and the only formula I've been taught is the one I stated, so I don't know any other ways of solving the problem.

bubokribuck · 2012-04-19 05:59:56

Just another question, if I'm only restricted to the formula stated earlier, is there any other way to solve the problem?

bobbym · 2012-04-19 07:52:57

Hi;

You mean the one in post #18? If you are restricted to that I do not see how to solve the problem.

bobbym · 2012-04-19 10:39:45

Hi;

This may not be the modified Euler method you want but there are a few different ones and this is the only one I could find that was useable on a system of equations.

We have

then I use the simple algorithm

pnew= pold+ h g(xold,yold,pold)
ynew= yold+ h f(xold,yold,pold)
xnew= xold+ h.

Here is how it works:

pold=1/10;
h=1/10;
yold=1/10;
xold=0;

For the numerics:

So y(.1) ≈ 0.11098.

Math Is Fun Forum

#1 2012-04-16 23:25:58

Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

#2 2012-04-17 00:25:28

Re: Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

#3 2012-04-17 02:31:05

Re: Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

#4 2012-04-17 05:03:12

Re: Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

#5 2012-04-17 07:56:39

Re: Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

#6 2012-04-17 08:00:04

Re: Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

#7 2012-04-17 10:33:54

Re: Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

#8 2012-04-17 10:48:21

Re: Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

#9 2012-04-17 11:39:28

Re: Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

#10 2012-04-17 11:46:18

Re: Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

#11 2012-04-17 11:53:59

Re: Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

#12 2012-04-17 12:07:33

Re: Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

#13 2012-04-17 12:15:25

Re: Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

#14 2012-04-17 12:41:20

Re: Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

#15 2012-04-17 13:38:20

Re: Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

#16 2012-04-17 22:02:27

Re: Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

#17 2012-04-17 22:04:00

Re: Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

#18 2012-04-18 04:33:18

Re: Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

#19 2012-04-18 07:07:05

Re: Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

#20 2012-04-18 12:23:35

Re: Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

#21 2012-04-18 18:46:38

Re: Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

#22 2012-04-19 05:58:23

Re: Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

#23 2012-04-19 05:59:56

Re: Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

#24 2012-04-19 07:52:57

Re: Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

#25 2012-04-19 10:39:45

Re: Rewrite y''+0.2y'(1-y^2)+y=0 as 1st order differential equation.

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